The Spear can't be stuck in Eva-01. We saw at the end of 2.0 that the Spear of Cassius is so powerful that it completely reverted Eva-01 to normal and put it into suspended animation. However, at the beginning of 3.0, Eva-01 temporarily activates and uses a laser attack on the Mark 4B. Later, Eva-01 is put in place as the Wunder's power reactor. Both of these things would have been impossible if the Spear were still in her.

Kaworu expected to find the Spear of Cassius down in Central Dogma. While Kaworu's dialogue is not always straightforward, and it's quite possible he's misleading Shinji in certain ways, he is also the only source for certain kinds of exposition we currently have. Thus, taking what Kaworu says at face value unless it's firmly contradicted is probably the best move for now. So we must assume that the Spear of Cassius was removed from Eva-01 so it could be used against the Lilith/Mark.06/12th Angel amalgamation, and it later changed into a Spear of Longinus for reasons we can only speculate upon.